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Mathematics of Computation

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On generalized averaged Gaussian formulas. II

Author: Miodrag M. Spalević
Journal: Math. Comp. 86 (2017), 1877-1885
MSC (2010): Primary 65D30, 65D32; Secondary 41A55
Published electronically: November 8, 2016
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Abstract: Recently, by following the results on characterization of positive quadrature formulae by Peherstorfer, we proposed a new $ (2\ell +1)$-point quadrature rule $ \widehat G_{2\ell +1}$, referred to as a generalized averaged Gaussian quadrature rule. This rule has $ 2\ell +1$ nodes and the nodes of the corresponding Gauss rule $ G_\ell $ with $ \ell $ nodes form a subset. This is similar to the situation for the $ (2\ell +1)$-point Gauss-Kronrod rule $ H_{2\ell +1}$ associated with $ G_\ell $. An attractive feature of $ \widehat G_{2\ell +1}$ is that it exists also when $ H_{2\ell +1}$ does not. The numerical construction, on the basis of recently proposed effective numerical procedures, of $ \widehat G_{2\ell +1}$ is simpler than the construction of $ H_{2\ell +1}$. A disadvantage might be that the algebraic degree of precision of $ \widehat G_{2\ell +1}$ is $ 2\ell +2$, while the one of $ H_{2\ell +1}$ is $ 3\ell +1$. Consider a (nonnegative) measure $ d\sigma $ with support in the bounded interval $ [a,b]$ such that the respective orthogonal polynomials, above a specific index $ r$, satisfy a three-term recurrence relation with constant coefficients. For $ \ell \ge 2r-1$, we show that $ \widehat G_{2\ell +1}$ has algebraic degree of precision at least $ 3\ell +1$, and therefore it is in fact $ H_{2\ell +1}$ associated with $ G_\ell $. We derive some interesting equalities for the corresponding orthogonal polynomials.

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Additional Information

Miodrag M. Spalević
Affiliation: Department of Mathematics, University of Beograd, Faculty of Mechanical Engineering, Kraljice Marije 16, 11120 Belgrade 35, Serbia

Keywords: Gauss quadrature, Gauss-Kronrod quadrature, averaged Gauss rules
Received by editor(s): February 13, 2016
Published electronically: November 8, 2016
Additional Notes: The author was supported in part by the Serbian Ministry of Science and Technological Development
Article copyright: © Copyright 2016 American Mathematical Society